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A338131 Triangle read by rows, T(n, k) = k^(n - k) + Sum_{i = 1..n-k} k^(n - k - i)*2^(i - 1), for 0 <= k <= n.

%I A338131 #18 Oct 30 2020 02:32:12
%S A338131 1,1,1,2,2,1,4,4,3,1,8,8,8,4,1,16,16,20,14,5,1,32,32,48,46,22,6,1,64,
%T A338131 64,112,146,92,32,7,1,128,128,256,454,376,164,44,8,1,256,256,576,1394,
%U A338131 1520,828,268,58,9,1,512,512,1280,4246,6112,4156,1616,410,74,10,1
%N A338131 Triangle read by rows, T(n, k) = k^(n - k) + Sum_{i = 1..n-k} k^(n - k - i)*2^(i - 1), for 0 <= k <= n.
%C A338131 This number triangle is case s = 2 of the triangles T(s; n,k) depending on some fixed integer s. Here are several (generalized) formulas and properties (attention: negative values are possible if s < 0):
%C A338131   (1) T(s; n,k) = k^(n-k) + Sum_{i=1..n-k} k^(n-k-i)*s^(i-1) for 0<=k<=n;
%C A338131   (2) T(s; n,n) = 1 for n >= 0, and T(s; n,n-1) = n for n > 0;
%C A338131   (3) T(s; n+1,k) = k * T(s; n,k) + s^(n-k) for 0<=k<=n;
%C A338131   (4) T(s; n,k) = (k+s) * T(s; n-1,k) - s*k * T(s; n-2,k) for 0<=k<=n-2;
%C A338131   (5) G.f. of column k: Sum_{n>=k} T(s; n,k)*t^n = ((1-(s-1)*t)/(1-s*t))
%C A338131       *(t^k/(1-k*t)) when t^k/(1-k*t) is g.f. of column k>=0 of A004248.
%F A338131 T(n,k) = ((k-1) * k^(n-k) - 2^(n-k)) / (k-2) if k <> 2, and T(n,2) = n * 2^(n-3) for n >= k.
%F A338131 T(n,n) = 1 for n >= 0, and T(n,n-1) = n for n > 0.
%F A338131 T(n+1,k) = k * T(n,k) + 2^(n-k) for 0 <= k <= n.
%F A338131 T(n,k) = (k+2) * T(n-1,k) - 2*k * T(n-2,k) for 0 <= k <= n-2.
%F A338131 T(n,k) = k * T(n-1,k) + T(n-1,k-1) - (k-1) * T(n-2,k-1) for 0 < k < n.
%F A338131 G.f. of column k >= 0: Sum_{n>=k} T(n,k) * t^n = ((1-t) / (1-2*t)) * (t^k / (1-k*t)) when t^k / (1-k*t) is g. f. of column k of A004248.
%F A338131 G.f.: Sum_{n>=0, k=0..n} T(n,k) * x^k * t^n = ((1-t) / (1-2*t)) * (Sum_{k>=0} (x*t)^k / (1-k*t)).
%e A338131 The number triangle T(n, k) for 0 <= k <= n starts:
%e A338131 n\ k :    0     1      2      3      4      5      6     7    8    9   10
%e A338131 =========================================================================
%e A338131    0 :    1
%e A338131    1 :    1     1
%e A338131    2 :    2     2      1
%e A338131    3 :    4     4      3      1
%e A338131    4 :    8     8      8      4      1
%e A338131    5 :   16    16     20     14      5      1
%e A338131    6 :   32    32     48     46     22      6      1
%e A338131    7 :   64    64    112    146     92     32      7     1
%e A338131    8 :  128   128    256    454    376    164     44     8    1
%e A338131    9 :  256   256    576   1394   1520    828    268    58    9    1
%e A338131   10 :  512   512   1280   4246   6112   4156   1616   410   74   10   1
%p A338131 T := proc(n, k) if k = 0 then `if`(n = 0, 1, 2^(n-1)) elif k = 2 then n*2^(n-3)
%p A338131 else (k^(n-k)*(1-k) + 2^(n-k))/(2-k) fi end:
%p A338131 seq(seq(T(n, k), k=0..n), n=0..10); # _Peter Luschny_, Oct 29 2020
%o A338131 (PARI) T(n,k) = k^(n-k) + sum(i=1, n-k, k^(n-k-i) * 2^(i-1));
%o A338131 matrix(7,7, n, k, if(n>=k, T(n-1,k-1), 0)) \\ to see the triangle \\ _Michel Marcus_, Oct 12 2020
%Y A338131 Cf. A004248.
%Y A338131 For columns k = 0, 1, 2, 3, 4 see A011782, A000079, A001792, A027649, A010036 respectively.
%K A338131 nonn,easy,tabl
%O A338131 0,4
%A A338131 _Werner Schulte_, Oct 11 2020